\documentclass[11pt,oneside]{amsart}
\usepackage{amssymb, amsmath, amsthm}

\usepackage{epsfig}
\usepackage{epstopdf}
\usepackage{graphicx}
\usepackage{color}
%\usepackage[shortalphabetic]{amsrefs}
\usepackage{mathptmx}
%\usepackage{hyperref}
%\usepackage{url}

\usepackage[normalem]{ulem}

\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}

%\usepackage{pinlabel}

%Define theorem styles
     \theoremstyle{definition}
      \newtheorem{theorem}{Theorem}[section]
      \newtheorem*{theorem*}{Theorem}
      \newtheorem{definition}[theorem]{Definition}
      \newtheorem{exercise}[theorem]{Exercise}
      \newtheorem{example}[theorem]{Example}
      \newtheorem{conj}{Conjecture}
      \newtheorem{proposition}[theorem]{Proposition}
      \newtheorem{corollary}[theorem]{Corollary}
      \newtheorem{claim}{Claim}
      \newtheorem{lemma}[theorem]{Lemma}
    
   % Define new commands

      \newcommand{\R}{{\mathbb R}}
      \newcommand{\C}{\mathbb C}
      \newcommand{\Z}{\mathbb Z}
	\newcommand{\N}{\mathbb N}
	\newcommand{\Q}{\mathbb Q}
      \newcommand{\nil}{\varnothing}
	\newcommand{\til}{\tilde}
	\newcommand{\boundary}{\partial}
	\newcommand{\m}{\backslash}
	\newcommand{\mc}[1]{\mathcal{#1}}
	\newcommand{\vect}[1]{\mathbf{#1}}
	\newcommand{\prob}[1]{\textbf{Problem {#1}: }}
	\newcommand{\defn}[1]{\textbf{#1}}
	\newcommand{\ob}[1]{\overline{#1}}
	%\newcommand{\ker}{\operatorname{ker}}
	\newcommand{\im}{\operatorname{im}}
	\newcommand{\tr}{\operatorname{tr}}
	\newcommand{\rank}{\operatorname{rank}}
	\newcommand{\rref}{\operatorname{rref}}
	
	\newcommand{\diver}{\operatorname{div}}
	\newcommand{\curl}{\operatorname{curl}}
	\newcommand{\grad}{\operatorname{grad}}

	\newcommand{\lpartial}[1]{\frac{\partial}{\partial #1}}
	\newcommand{\llpartial}[2]{\frac{\partial #1}{\partial #2}}
	\newcommand{\co}{\mskip0.5mu\colon\thinspace} 
	\newcommand{\spacing}{\parskip 6.6pt
\parindent 0pt}

	
	
	
	\newenvironment{enumer}
		{\begin{enumerate}
		\parskip 6.6pt
		\parindent 0pt
		\renewcommand{\labelenumi}{(\alph{enumi})}
		
		}
		{\end{enumerate}}


	  
\parskip 6.6pt
\parindent 0pt

%\renewcommand{\labelenumi}{(\alph{enumi})}


	
   % The following mysterious maneuver gets rid of AMS junk at the top
   % and bottom of the first page.
   
      \makeatletter
      \def\@setcopyright{}
      \def\serieslogo@{}
      \makeatother
   
% This ends the preamble.  We now proceed to the document itself.

\begin{document}


   % title

   \title[short title]{long title}
   \author{}
   % Note that the short title for running heads goes in square
   % brackets.  This is optional.  The long title goes in curly
   % braces.  In the long title, line breaks are indicated by \\.


%Here are two sets of commands for including images.
  
%\begin{center}
%\begin{figure}[h]
%\includegraphics[scale=0.4]{FinalExam-prob10.pdf}
%\end{figure}
%\end{center}

%\begin{figure}[ht] 
%\scalebox{0.5}{\input{Pen.eps_t}}
%\caption{A pox on both your houses!}
%\end{figure}

   

   % today's date, or fill in whatever date you prefer
   \date{\today}

\framebox{
\raisebox{0in}[.5in][.2in]{
\textbf{MA 253 Homework Problems 6} 
}
}



Homework has both a ``Reading'' portion and a ``Problems'' portion. It is essential that you do the reading by the next class. The reading assignments are posted on a separate webpage. Don't forget to do them!

All page numbers and section numbers refer to the 5th edition of Bretscher's \uline{Linear Algebra}. Note that most odd numbered problems have solutions in the back of the text. Problems without solutions are worth more points than those with solutions.

Remember to use \emph{Mathematica} to do row operations!

 \begin{enumerate}\spacing
 
 \item Suppose that $W, V \subset \R^n$ are subspaces of $\R^n$ with $W \subset V$ and that $\dim W = \dim V - 1$. Suppose that $\vect{w}_1, \hdots, \vect{w}_k$ is a basis for $W$. Explain why if $\vect{v}$ is any vector not in $W$, then $\vect{w}_1, \hdots, \vect{w}_k, \vect{v}$ is a basis for $V$.

\item Explain why every subspace of $\R^n$, other than $\{\vect{0}\}$ has a basis.

 \item \uline{Section 3.3 (page 131)} \newline
  
  \begin{enumerate}\spacing

\item Problems 62, 63, 67, 68

 \end{enumerate}

  \item \uline{Section 3.4 (page 159)} \newline
\begin{enumerate}\spacing
\item Problems 4 - 11 (Remember to use mathematica to do row row reductions, invert matrices, etc.)
\item Do parts (a) and (b) of Problems 21 - 24
\item Problems 39 - 42, 57, 71
\end{enumerate}

\item Use change of basis methods to find the matrix (in standard coordinates) for the following linear transformations:
\begin{enumerate}\spacing
\item Rotation by an angle of $\theta$ around the line in $\R^3$ spanned by the vector $(2, -2, 7)$.
\item The linear transformation $\R^3 \to \R^3$ which is first reflection across the plane with equation $2x + y + z = 0$ and then reflection across the plane $x - y - z = 0$.
\item The linear transformation $\R^4 \to \R^4$ which is a reflection across the 3-dimensional subspace defined by the equation $2x + y - z + w = 0$. 
\item We can think of points of $\R^4$ as points of  the form $((a,b), (c,d))$ where $(a,b) \in \R^2$ and $(c,d) \in \R^2$. Let $T$ by the linear transformation of $\R^4$ which reflects points in the first copy of $\R^2$ across the plane defined by $x + y + z = 0$ and which simultaneously rotates the points in the second copy of $\R^2$ by an angle of $\pi/6$ around the vector $(4,1)$.
\end{enumerate}

\item Suppose that $T \co \R^n \to \R^n$ is a linear transformation such that there is a basis $\mc{B} = \{\vect{b}_1, \hdots, \vect{b}_n\}$ such that the matrix $[T]_\mc{B}$ is diagonal. Explain (geometrically) what the linear transformation is doing.

\item To get some feel for how one might develop intuition for working in $\R^4$ watch the promotional video for a videogame under construction called \emph{Miegakure}. You can find it on Youtube: 

\href{https://www.youtube.com/watch?v=9yW--eQaA2I#t=16}{https://www.youtube.com/watch?v=9yW--eQaA2I#t=16} 

or as the second video (scroll down) on the games website: 

\href{http://miegakure.com/}{http://miegakure.com/}. 

Another great introduction to thinking about the 4th dimension is the classic novel Flatland (which you can find online or in the library), but you don't have to read that if you don't want to.


\end{enumerate}



\end{document}